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DLR-IB-RM-OP-2017-164 System Identification and Parameter Space Control Design for a Small Unmanned Aircraft Bachelor’s Thesis

Andre Fialho Coelho

André Fialho Coelho

System Identification and Parameter SpaceControl Design for a Small Unmanned Aircraft

Campos dos Goytacazes – Brazil

2017

André Fialho Coelho

System Identification and Parameter Space ControlDesign for a Small Unmanned Aircraft

Thesis presented to the Instituto Federal Flu-minense, Brazil as a partial requisite to theawarding of the five-year Bachelor degree inControl and Automation Engineering.

Instituto Federal de Educação, Ciência e Tecnologia Fluminense - IFF Campos – Centro

Control and Automation Engineering Program

Supervisor: Dr. Alexandre Carvalho LeiteCo-supervisor: Tin Muskardin (German Aerospace Center – DLR)

Campos dos Goytacazes – Brazil2017

Acknowledgements

I am grateful to God for having helped me in all moments. Without Him nothingwould have been possible.

I thank the Instituto Federal Fluminense for having provided me with the meansto become a control and automation engineer.

I would also like to express my gratitude to my supervisors, Prof. Alexandre Leite,for teaching me a little about his way of thinking and solving problems, which I havealways admired. And Tin Muskardin, for always being solicitous and willing to teach andhelp, characteristics that make him an admirable engineer and researcher.

I am especially grateful to my family for all love and support. For providing mewith a home where I could grow personally and professionally.

A very special gratitude goes to my future wife, Raquel, for always being there forme and unconditionally supporting me in my dream of becoming a good engineer.

AbstractInitial tests in cooperative control for autonomous landings of an Unmanned Aerial Vehicle(UAV) on a moving car have presented promising results. However, the identification of ahigh-fidelity simulation model is a step of great importance towards the development ofmore effective model predictive control strategies, which rely on precise models to allowcooperative control of High Altitude, Long Endurance (HALE) UAVs with autonomousground vehicles. In this context, this work aims to develop a reliable model for the PenguinBE aircraft used in cooperative landing tests at the German Aerospace Center (DLR),and to design a high-performance pitch attitude controller applying the parameter spaceapproach. The system identification procedure has been carried out by applying both, theOutput Error and the Two Step methods, and a linear longitudinal model of the aircrafthas been developed. Parameter Space Control has been applied to the identified model inorder to suggest a set of alternative gains to the ones that are currently in use, which havebeen fine-tuned in flight, using the Ziegler-Nichols method, which will not be viable in thescope of stratospheric missions.

Keywords: System identification. Parameter space control design. Output Error Method.Two Step Method. UAV.

List of Figures

Figure 1 – Successful cooperative landing with demonstrator setup. . . . . . . . . 10Figure 2 – Elektra One Solar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Figure 3 – Penguin BE UAV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Figure 4 – Simulink R© model of Penguin BE, created using AeroSim Blockset. . . . 13Figure 5 – Example of DLR 3211 input. . . . . . . . . . . . . . . . . . . . . . . . 15Figure 6 – Block schematic of the Output Error method. . . . . . . . . . . . . . . 17Figure 7 – Details of the OEM-Software. . . . . . . . . . . . . . . . . . . . . . . . 18Figure 8 – Simplified aircraft block diagram. . . . . . . . . . . . . . . . . . . . . . 22Figure 9 – Optimal inputs in the time and frequency domains. . . . . . . . . . . . 28Figure 10 – Actual flight data inputs in the time and frequency domains. . . . . . . 29Figure 11 – Bode plots of derivative terms. The blue, red, and black dashed rectan-

gles represent the regions of identifiability of the phugoid, short period,and throttle doublet maneuvers, respectively. . . . . . . . . . . . . . . . 32

Figure 12 – Inputs used in identification. . . . . . . . . . . . . . . . . . . . . . . . . 34Figure 13 – Inputs used in validation. . . . . . . . . . . . . . . . . . . . . . . . . . 35Figure 14 – States simulated with the a priori model and measured states. . . . . . 36Figure 15 – States simulated with the model resulting from the Output Error Method

and measured states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Figure 16 – Measured longitudinal and lateral states and states generated from the

integration of the 6DOF equations of motion with estimated biases added. 40Figure 17 – Reconstructed forces and moments, and forces and moments from linear-

regression – validation set. . . . . . . . . . . . . . . . . . . . . . . . . . 42Figure 18 – Measured and Two Step Method simulated states. . . . . . . . . . . . . 43Figure 19 – Measured States and states resulting from the Output Error and Two

Step identified models. . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Figure 20 – System performance in the parameter space. All specification regions on

the left. Intersection of specifications on the right. Left-hand-side plots:yellow – stable system; blue – overshoot less than 5%; green – settlingtime less than 10 seconds; red – rise time less than 1 second; black –gains currently in use. Right-hand-side plots: black – gains where allspecifications are met at the same time; green – gains currently in use. 49

Figure 21 – Intersection of specifications in the three-dimensional parameter space.Black spheres – gains where all specifications are met at the same time;green sphere – gains currently in use . . . . . . . . . . . . . . . . . . . 50

List of Tables

Table 1 – Characteristics of the model validation statistic measures . . . . . . . . 24Table 2 – Values tuned in flight and grid search range of gains . . . . . . . . . . . 26Table 3 – Characteristics of the the optimal elevator and throttle inputs. . . . . . 27Table 4 – Characteristics of the inputs from flight data . . . . . . . . . . . . . . . 29Table 5 – Identifiability of derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 33Table 6 – Statistical measures of the states simulated using the a priori model –

identification set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Table 7 – Statistical measures of the states simulated using the a priori model –

validation set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Table 8 – Statistical measures of the states resulting from the Output Error Method

– identification set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Table 9 – Statistical measures of the states resulting from the Output Error Method

– validation set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Table 10 – Estimated measurement biases for each maneuver . . . . . . . . . . . . 39Table 11 – Statistical measures of the reconstructed states – validation set . . . . . 40Table 12 – Linear regression terms and respective values . . . . . . . . . . . . . . . 41Table 13 – Statistical measures of the Two Step Method estimated forces and mo-

ments – identification set . . . . . . . . . . . . . . . . . . . . . . . . . . 41Table 14 – Statistical measures of the Two Step Method estimated forces and mo-

ments – validation set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Table 15 – Statistical measures of the Two Step Method forces and moments ob-

tained by Lee (2017) – identification set . . . . . . . . . . . . . . . . . . 41Table 16 – Statistical measures of the Two Step Method forces and moments ob-

tained by Lee (2017) – validation set . . . . . . . . . . . . . . . . . . . . 42Table 17 – Statistical measures of the states resulting from the Two Step Method –

identification set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Table 18 – Statistical measures of the states resulting from the Two Step Method –

validation set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Table 19 – Statistical measures of the Two Step Method states obtained by Lee

(2017) – identification set . . . . . . . . . . . . . . . . . . . . . . . . . . 44Table 20 – Statistical measures of the Two Step Method states obtained by Lee

(2017) – validation set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Table 21 – Performance indicators of the actual control system . . . . . . . . . . . 47Table 22 – Definition of the concise derivatives . . . . . . . . . . . . . . . . . . . . 57

List of symbols

u, v, w Linear velocities along the x, y, and z axes, respectively, in the bodyframe

ax, ay, az Linear accelerations along the x, y, and z axes, respectively, in the bodyframe

φ, θ, ψ Roll, pitch and yaw angles, respectively

p, q, r Roll, pitch and yaw rates, respectively

ω Engine angular velocity

h Aircraft altitude

X,Z Axial and normal forces, respectively

M Pitching moment

Contents

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.1 Previous work on the application . . . . . . . . . . . . . . . . . . . . . 91.2 Penguin BE UAV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 IDENTIFICATION AND CONTROL DESIGN FOR PENGUIN BE . 122.1 A priori model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Optimal Input Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Output Error Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.1 Maximum likelihood estimation and cost function formulation . . . . . . . 182.4 Two Step Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4.1 Flight path reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . 202.4.2 Estimation of the force and moment components . . . . . . . . . . . . . . 212.5 Model validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.6 Parameter space control design . . . . . . . . . . . . . . . . . . . . . 24

3 RESULTS AND DISCUSSION . . . . . . . . . . . . . . . . . . . . . 273.1 Considerations about the experiment . . . . . . . . . . . . . . . . . . 273.2 System identification results . . . . . . . . . . . . . . . . . . . . . . . 353.2.1 Results of the a priori model . . . . . . . . . . . . . . . . . . . . . . . . . 353.2.2 Results of the Output Error Method . . . . . . . . . . . . . . . . . . . . . 363.2.3 Results of the Two Step Method . . . . . . . . . . . . . . . . . . . . . . . 393.2.3.1 Flight path reconstruction results . . . . . . . . . . . . . . . . . . . . . . . . 393.2.3.2 Results of force and moment components identification . . . . . . . . . . . . . 403.2.4 Comparison between identification methods . . . . . . . . . . . . . . . . . 453.3 Results of Parameter Space Control Design . . . . . . . . . . . . . . 47

4 CONCLUSION AND OUTLOOK . . . . . . . . . . . . . . . . . . . 51

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

APPENDIX 56

APPENDIX A – CONCISE LONGITUDINAL AERODYNAMIC DERIVA-TIVES AND ENGINE DYNAMICS . . . . . . . . 57

8

1 INTRODUCTION

Recent studies being performed at the German Aerospace Center (DLR) aimto develop new technologies in order to land High Altitude, Long Endurance (HALE)Unmanned Aerial Vehicles (UAVs) on a platform mounted on the top of a ground vehicle,which, cooperatively with the UAV, controls position and velocity until successful landingis achieved. The main purpose is to eliminate the need for an aircraft-mounted landinggear and also to facilitate landings in crosswind situations. As a result, the total massof the aircraft would be significantly reduced. This would allow its operation for longperiods of time since this kind of UAVs demands very lightweight structures and at thesame time heavy batteries, required for overnight flight. If a reliable integrated landingsystem for such UAVs is successfully developed, a large step will be taken towards theircommercial use. The main advantage of such ultralight stratospheric aircraft consists inits versatility to be applied in numerous applications, including tasks that are usuallyperformed by satellites, e.g. earth observation, atmospheric research, and communicationnetworks. HALE UAVs are attractive options to supplement or substitute satellites dueto high costs, necessity of rocket launch, and dependence of orbits intricate to the use ofthese spacecraft.

Cooperative landings have successfully been performed with the use of a demon-strator setup (Fig. 1), comprised of a Penguin BE small unmanned aircraft and a car witha human driver. However, a high fidelity model of the small UAV is required in orderto eliminate or reduce the need for in-flight tuning of control parameters and to furtherimprove the performance of the system through optimal control design. Adding to that, areliable simulation model of the aircraft would accelerate the process of developing moreeffective cooperative control strategies to finally allow the first tests with actual HALEUAVs and unmanned ground vehicles. The approach that will be applied, namely ModelPredictive Control, aims to generate feasible and optimal trajectories, based on a prioriknowledge of the vehicles. Therefore, deficient models would compromise the applicationof such technique.

In this context, this work aims to develop a reliable model for the Penguin BEaircraft used in cooperative landing tests, as well as to analyze the pitch attitude controldesign that is currently in use and suggest alternative designs within the predefinedspecifications.

Two different approaches are applied in system identification, namely time-domainOutput Error Method and Two Step Method. Control design is performed through theapplication of a parameter space analysis. The thesis is divided into two major parts:

Chapter 1. INTRODUCTION 9

Chapter 2 provides an explanation of the identification methods being utilized and howthey have been applied, as well an overview of the model validation tools adopted. Italso provides a presentation of Parameter Space Control Design and the way it has beenadapted for the intended application. Furthermore, related papers on each subtopic arealso cited in this chapter. In sequence, Chapter 3 presents the results of the identificationmethods and a comparison between them, as well as the results of control design andanalysis in the space of parameters.

1.1 Previous work on the applicationSeveral articles and conference papers about the DLR cooperative landing of HALE

UAVs have been published. The concept was firstly introduced by Laiacker et al. (2013),which is mainly focused on the presentation of the multi-sensor system used in the landings.The paper gives specific details about vision-based state estimation and sensor data fusionmethods employed.

A couple of years later, Muskardin et al. (2016) presented details about the cooper-ative control algorithm and the results of the first simulations. Subsequent developmentand the first results of successful landings with the demonstrator setup are presented byMuskardin et al. (2016). Ultimately, Muskardin et al. (2017) also adds new advances withinthe application and results of additional landing experiments. The main improvements arethe inclusion of vision-based state estimation and some alterations in the the high-levelmission control structure in order to provide safer and smoother cooperative landings.

Adding to that, three master’s theses have been produced within the field. Thefirst, Balmer (2015), was mainly concerned about modeling, simulation, and developmentand testing of aircraft control strategies, including Total Energy Control System (TECS)(KASTNER; LOOYE, 2013). The second, Persson (2016), performed a linear analysisof the cooperative system, further developed control strategies, and presented resultsof the first tests. The third, Lee (2017), aimed to identify high fidelity models for bothPenguin BE and the optionally piloted UAV Elektra One Solar1 (Figure 2). The resultsand methods applied to Penguin BE system identification are compared with the onesachieved in this work.

1.2 Penguin BE UAVThe aircraft being used in the demonstrator setup, whose longitudinal model is

identified in this work, is a Penguin BE UAV (Fig. 3), developed by UAV Factory2. It is an1 <https://www.elektra-solar.com/>2 <http://www.uavfactory.com/>

https://www.elektra-solar.com/

http://www.uavfactory.com/


Figure 1 – Successful cooperative landing with demonstrator setup.

Source: Muskardin et al. (2017).

Figure 2 – Elektra One Solar.

Source: Lee (2017).

electric high-wing unmanned aerial vehicle with a negative V-tail configuration, poweredby a 640 Wh battery cartridge made from 48 lithium-polymer cells. Brsides that, thisaircraft is propelled by a geared brushless DC motor and a 19×11 inch propeller in pusherconfiguration. Furthermore, it has 3.3 meters of wing span and 2.27 meters of length. Itsempty weight, including battery cartridge and standard landing gear, is 14.9 kg with amaximum payload capacity of 6.6 kg (UAV FACTORY).


Figure 3 – Penguin BE UAV.

Source: UAV Factory.

The DLR’s Penguin BE UAV instrumentation provides measurements of thevariables used in system identification. A LORD MicroStrain3 3DM-Gx3-25 inertial mea-surement unit (IMU) measures linear accelerations, angular velocities and orientation, ata sampling rate of 100 Hz. The Swiss Air-Data4 PSS-8 pitot-static system, mounted onthe nose of the aircraft, provides true airspeed and temperature data, at a 20 Hz samplingrate. The Novatel5 Flex-Pak6 RTK receiver comprises a differential GPS and is responsiblefor measuring latitude, longitude and altitude, at a sampling rate of 20 Hz. In addition, awireless local-area network (WLAN) sets up the communication with the ground station.

3 <http://www.microstrain.com/>4 <http://www.swiss-airdata.com/>5 <http://www.novatel.com/>

http://www.microstrain.com/

http://www.swiss-airdata.com/

http://www.novatel.com/

12

2 IDENTIFICATION AND CONTROL DE-SIGN FOR PENGUIN BE

2.1 A priori modelThe first attempt to model and simulate the DLR’s Penguin BE aircraft was

performed by Balmer (2015). In his master’s thesis, Balmer developed a Simulink R© modelof the aircraft, which was created using the AeroSim Blockset (UNMANNED DYNAMICS).In order to model the aerodynamics, an integration of the potential flow solver namedAthena Vortex Lattice (AVL) (DRELA; YOUNGREN, 2004) and a program for thedesign and analysis of subsonic isolated airfoils, XFOIL (DRELA; YOUNGREN, 2000),was performed, according to the method described in Klöckner (2013). Furthermore, apropulsion model was developed from both, experimental data and information provided bythe manufacturer. In order to estimate the total moment of inertia, the aircraft componentswere modeled as hollow hulls (fuselage), flat plates (wings and tail), or point masses (otherparts). Then the aircraft states have been calculated by introducing the modeled forcesand moments into the six-degree-of-freedom (6DOF) nonlinear equations of motion. Anoverview of the resulting model can be seen in Figure 4.

As mentioned by Balmer (2015), the conceived model presented inaccuracies whichcompromised its use in inner-loop control design; therefore control parameters had to bere-tuned in flight by applying the Ziegler-Nichols method.

Despite the mentioned inaccuracies, a work developed by Persson (2016) describedan effort to obtain a linear model of the Penguin BE UAV through trim and linearization ofthe Simulink model presented by Balmer (2015). The resulting state-space model describingthe longitudinal dynamics of the aircraft is given in Equation 2.1.

u

w

q

θ

ω

=

−0.10 0.39 −1.4 −9.8 0.006−0.64 −3.6 22 −0.6 00.19 −2.8 −5.6 0 −0.001

0 0 1 0 021 1.2 0 0 −2.6

u

w

q

θ

ω

+

0.38 0−7.3 0−65 0

0 00 2027

δeδt

(2.1)

where u and w are the linear velocities along x and z axes, respectively, in the body frame;q corresponds to the pitch rate; θ denotes the pitch angle; and ω represents the engineangular velocity. δe and δt are the elevator and throttle commands, respectively. All statesand measurements are in the International System of Units (SI).

Chapter 2. IDENTIFICATION AND CONTROL DESIGN FOR PENGUIN BE 13

Figure 4 – Simulink R© model of Penguin BE, created using AeroSim Blockset.

Source: Balmer (2015).

The system has a total of five poles, with one real pole from engine dynamics andtwo complex-conjugate pairs related to the phugoid and short period modes, described inSection 2.2.

Phugoid: -0.03072 +− 0.5349jShort period: -4.5935 +− 7.7958jEngine dynamics: -2.6515

Being the most accurate linear representation of the aircraft available, this setuphas been used in this work as a priori model.

2.2 Optimal Input DesignThe optimal input design process consists of defining a set of maneuvers, based on

a priori knowledge of the system, which best excite the aircraft modes, in an attempt tomaximize the effect of their dynamics in the measurements of the states. This step is crucialto aircraft system identification since the estimation of a given parameter is impossibleunless its variation affects the data being analyzed. Therefore, the accuracy and reliability


of estimates are closely attached to how much information about the desired parameter isavailable in the measured response. This amount of information, in turn, clearly dependson the nature of the given input for linear and nonlinear dynamical systems, includingaircraft (SHANNON, 1948).

Since Milliken Jr. (1947), the design of proper inputs has been widely consideredas a step of great importance for aircraft parameter estimation purposes. Much work hasbeen performed in the area up to date. Mehra (1974), Hamel and Jategaonkar (1996)present reviews on the topic. Qian, Nadri and Dufour (2017) is a more recent study onthe application, focused on nonlinear system identification.

According to Jategaonkar (2015), two approaches are commonly applied to optimalmaneuver design. The first is based on statistical properties of the estimate and is performedby assuming that the amount of information about a specific parameter in the data beinganalyzed is a function of the overall model and of the inputs given. Therefore, by keepingthe parameters fixed at some a priori values, the variation of the information contentwill be mainly determined by the inputs applied to the system. That being performed, anoptimization procedure, having input shape as parameter, is carried out with the intent ofachieving the maximum amount of information. This is usually determined by the Fishermatrix (FISHER, 1934), given by Equation 2.2.

Fij = E

{∂2 ln p(z | Θ)∂Θi∂Θj

}(2.2)

where p(z | Θ) represents the conditional probability density function of measurementsz with respect to a parameter vector Θ, and E{.} denotes the expected value. A moredetailed view on the statistical approach can be found in Gupta and Hall Jr. (1975), whereboth, time and frequency domain input designs are demonstrated.

The second approach, which is used in this work, is an engineering approach basedon the frequency of the aircraft eigenmodes. In this method, the natural frequencies of ana priori model of the aircraft are computed, and input signals are conveniently designedto exhibit maximum energy around those frequencies. This approach is based on theprinciple that deterministic signals can be represented by a linear combination of complexexponentials in the time domain, which represent components with a specific amplitude at aspecific point in the frequency domain. This is achieved by computing the Fourier Transformof the signals (FOURIER, 1878). Therefore, a set of multistep inputs are convenientlydesigned to exhibit larger components in a sufficiently wide range of frequencies to excitethe natural vibrations of the a priori model. The use of multistep signals is also justifiedby their relative easiness to be applied manually. Other types of signals like the Mehrainput and the DUT input may also be used (PLAETSCHKE; SCHULZ, 1979). However,these approaches lead to continuous smoothly-varying maneuvers and usually require an


autopilot to be properly performed.

For the purpose of this work, only the longitudinal aircraft modes are to be excited,namely phugoid (or long period) and short period. The phugoid is a lightly dampedlow-frequency oscillation (generally between 0.1 rad/s and 1 rad/s) in the axial velocityu, which is also visible in pitch attitude θ, altitude h, and, to a lesser extent, in angle ofattack α and pitch rate q. The short period mode, in contrast, is a damped high-frequencyoscillation (typically between 1 rad/s and 10 rad/s) in the pitch attitude θ, which mainlyaffects pitch rate q and angle of attack α (COOK, 2012).

According to Jategaonkar (2015), since the short period mode is well damped, itrequires a fast changing input to excite it. The signal used to induce this oscillation wasthe so called DLR 3211 (KALETKA et al., 1989), which consists of a pulse with three timesteps (∆t) duration and amplitude of 0.8, another pulse with 2∆t and amplitude of -1.2,followed by one positive and one negative pulse of duration ∆t each and amplitude of 1.1.In this case, the amplitudes presented correspond to the ratio between the amplitude ofeach pulse and the average absolute value of all amplitudes in the maneuver. An exampleof the DLR 3211 input is shown in Figure 5.

Figure 5 – Example of DLR 3211 input.

Source: The author.

The phugoid mode, being lightly damped, can be satisfactorily excited by a pulsesignal. A suitable ∆t for the phugoid maneuver has been chosen such that its bandwidthis large enough to account for uncertainties in the a priori model and its energy is not


unnecessarily spread over an excessively large range of frequencies. In addition, a throttledoublet has also been used to excite the long period motion in order to identify the effectof the engine dynamics in the system. A doublet in thrust is usually used for this purpose;however a satisfactory model of the aircraft engine has not yet been obtained and therelationship between throttle and thrust has not been mapped yet. The time steps of thedoublet and DLR 3211 maneuvers have been calculated such that their energy peak is atthe natural frequency of the mode they are intended to excite. The formulas presented inJategaonkar (2015) to achieve optimal time steps for the doublet and DLR 3211 maneuversare presented in Equations 2.3 and 2.4, respectively.

∆tDBLT = 2.3ωn

(2.3)

∆t3211 = 1.6ωn

(2.4)

where ∆tDBLT and ∆t3211 denote the time steps of the doublet and DLR 3211 maneuvers,respectively, and ωn is the undamped natural frequency of the mode to be excited.

Jategaonkar (2015) also states that, for parameter identification, the system shouldbe allowed to oscillate freely for at least one period of oscillation of the intended motion.

The designed maneuvers have been used as inputs to the Simulink R© model presentedby Balmer (2015). Their amplitudes have been adjusted to avoid instability and actuatorsaturations.

Ultimately, Bode plots containing the frequency response of each term of the apriori model have been used to check if the set of designed maneuvers is sufficient toestimate all derivatives. “Term”, in this case, is used to denote the product of a concisederivative and a state, e.g. xuu. In this stage a rule of thumb, presented by Plaetschke andSchulz (1979) is used. This criterion states that a derivative is not considered identifiableunless its term represents at least 10% of the largest term’s magnitude in the maneuverfrequency range.

2.3 Output Error MethodAccording to Jategaonkar (2015), the Output Error Method (OEM) is the most

widely adopted approach in time-domain aircraft parameter estimation from measuredflight data (WANG; ILIFF, 2004; MORELLI; KLEIN et al., 2005). It consists of iterativelyadjusting model parameters in order to minimize the error between measured data, fromflight tests, and predicted data, from integration of state-space model equations.

In the application of this method, the state-space setup presented in Section 2.1has been adopted as a priori model. The derivatives considered identifiable according to


the 10% rule, presented in Section 2.2, have been estimated for a satisfactory fit betweenthe measured and simulated states. In addition, constant terms have been added to thestate equations in order to account for measurement biases. On that account, the systemhas been considered to be as shown in Equation 2.5.

x(t) =Ax(t) +Bu(t) + bx, x(t0) = x0

y(t) =Cx(t) + by(2.5)

where bx and by correspond to state and measurement bias vectors, respectively.

The Output Error system identification has been carried out through the applicationof the OEM-Software, presented by Jategaonkar (2015), with some minor modifications.Figures 6 and 7 display the block schematics of the Output Error Method and its imple-mentation within the OEM-Software, respectively.

Figure 6 – Block schematic of the Output Error method.

Source: Jategaonkar (2015).

The major steps taken in this work to achieve minimum output error followed theguidelines presented in Jategaonkar (2015), which are

1. Setting initial values of the parameters according to the a priori model.

2. Computing the outputs of the model through integration of the state equations, andcomparing them with measurements.

3. Estimating the measurement noise covariance matrix.

4. Minimizing the cost function defined in Equation 2.9.

5. Iterating on step two and checking for convergence.


Figure 7 – Details of the OEM-Software.

Source: Jategaonkar (2015).

The minimization of the cost function has been performed through the applicationof the Levenberg-Marquadt method (JATEGAONKAR, 2015). In addition, the stateequations have been numerically integrated by a fourth-order Runge-Kutta integrationmethod, which is included in the OEM-Software (JATEGAONKAR, 2015). Furthermore,the error between the measurements and model outputs has been minimized and optimalparameters have been obtained through maximum likelihood estimation.

2.3.1 Maximum likelihood estimation and cost function formulation

The maximum likelihood estimation is based on the maximization of the likelihoodfunction, defined in Equation 2.6 (FISHER, 1925).

p(z | Θ) =N∏k=1

p(zk | Θ) (2.6)


where p(z | Θ) is the probability of a set of N observations z, given the unknown parametervector Θ.

The procedure consists of finding suitable values of Θ that maximize p(z | Θ).Contrasting to what is performed in the statistical approach of optimal input design (seeSection 2.2), in the Output Error Method, inputs recorded during the flight test are appliedand model parameters are varied in order to maximize the likelihood function. For the sakeof the fact that p(z | Θ) is usually exponential, its negative logarithm is commonly utilizedto find optimal parameters, without affecting the solution. The expression of the logarithmof the probability density function p with respect to measurements z, parameters Θ, andthe measurement error covariance matrix, R, is given in Equation 2.7. (JATEGAONKAR,2015)

L(z | Θ, R) = 12

N∑k=1

([z(tk)− y(tk)]TR−1[z(tk)− y(tk)]+

N

2 ln [det(R)] + Nny2 ln (2π)

) (2.7)

where z(tk) and y(tk) represent the measurements and model outputs, respectively, andny is the number of system outputs.

The error covariance matrix, R, is defined in Equation 2.8.

R = 1N

N∑k=1

([z(tk)− y(tk)][z(tk)− y(tk)]T

)(2.8)

The set of parameters that ensure maximum likelihood between flight measurementsand model outputs result from the minimization of Equation 2.7. The cost function canbe simplified to Equation 2.9 (JATEGAONKAR, 2015).

J(Θ) = det(R) (2.9)

This definition of the cost function ensures the computational feasibility of themethod.

2.4 Two Step MethodThe Two Step Method is a system identification procedure comprised of two

stages, namely flight path reconstruction (LOMBAERTS et al., 2010; TEIXEIRA et al.,2011; MULDER et al., 1999) and estimation of force and moment parameters throughlinear regression. Its advantages are the possibility of estimating measurement biasesindependently from derivative terms, avoiding correlations between these quantities, the


elimination of the need for initial guesses for the parameters, and the assurance of globalminimum convergence within two iterations in the second stage. This approach has beenperformed by numerous authors, including Grymin and Farhood (2016), Grymin (2013),and Oliveira et al. (2005). Lee (2017) has performed two step system identification of thesame aircraft analyzed in this work; however a stochastic approach based on an ExtendedKalman Filter (EKF) has been applied in first step, rather than the deterministic one,based on maximum likelihood estimation, performed in this composition. Martin and Feik(1982), Evans et al. (1985), and Jategaonkar (2015) state that both procedures usuallyyield similar results; however the deterministic approach is more advantageous when thereis no accurate a priori knowledge about noise statistics.

2.4.1 Flight path reconstruction

Flight path reconstruction, also known as data compatibility check, consists inverifying the compatibility of measured data by inputting the variables recorded duringflight into the 6DOF rigid-body kinematic equations. This allows for the estimation ofinstrument errors present in the measurements. The procedure is also performed using theOEM-Software presented in Section 2.3.

In this stage both lateral and longitudinal measurements are used, although onlyestimates of errors present in the latter are adopted as inputs to the next step.

The nonlinear state equations used in this stage are shown in Equation 2.10.

u = −(qm −∆q)w + (rm −∆r)v − g sin θ + (aCGxm −∆ax) , u(t0) = u0

v = −(rm −∆r)u+ (pm −∆p)w − g cos θ sinφ+ (aCGym −∆ay) , v(t0) = v0

w = −(pm −∆p)v + (qm −∆q)u− g cos θ cosφ+ (aCGzm −∆az) , w(t0) = w0

φ = (pm −∆p) + (qm −∆q) sinφ tan θ + (rm −∆r) cosφ tan θ , φ(t0) = φ0

θ = (qm −∆q) cosφ− (rm −∆r) sinφ , θ(t0) = θ0

ψ = (qm −∆q) sinφ sec θ + (rm −∆r) cosφ sec θ , ψ(t0) = ψ0

h = u sin θ − v cos θ sinφ− w cos θ cosφ , h(t0) = h0

(2.10)

where v denotes the aircraft’s linear velocity along the y-axis in the body frame; φ and ψrepresent the roll and yaw angles; h stands for the aircraft altitude; pm, qm, and rm arethe measured angular rates; aCGxm , aCGym , and aCGzm are the measured linear accelerations atthe aircraft center of gravity; ∆ stands for biases in the measured quantities.

The OEM-Software compares the integrated state equations and calculates thebiases through maximum likelihood estimation (see Section 2.3). The procedure is runiteratively in order to minimize errors between the model outputs and measured data.A good match means that the estimated biases are similar to the ones present in the


measurements. This is performed through the minimization of the cost function presentedin Equation 2.9, with instrument biases as parameters. In this part, different biases foreach maneuver were considered, as suggested by Jategaonkar (2015).

Once the biases are estimated, the longitudinal forces and moments, which wereacting on the aircraft during the flight experiment, could be reconstructed from thecorrected linear accelerations and angular rates. Since the maneuvers performed weremeant to excite only the longitudinal dynamics, the estimates of biases in lateral motionare not reliable.

2.4.2 Estimation of the force and moment components

The second stage of the Two Step Method is a linear regression of the force andmoment equation parameters. After an estimate of the linear accelerations and angularrates has been obtained, the total longitudinal forces and moments (aerodynamic andpropulsion) have been reconstructed from the corrected measurements. However, in orderto obtain mathematical expressions for the forces and moments, it is necessary to discoverthe contributions of each state and input to the result of the previous step. In orderto accomplish that, the expressions for the total longitudinal forces and moments areconsidered to be a linear combinations of states and inputs, as shown in Equation 2.11.

X = X0 +Xuu+Xww +Xqq +Xδeδe+Xδtδt

Z = Z0 + Zuu+ Zww + Zqq + Zδeδe

M = M0 +Muu+Mww +Mqq +Mδeδe

(2.11)

where X, Z, and M correspond to the reconstructed axial and normal forces, and thepitching moment, respectively. The components that multiply inputs and states on theright-hand side are constant terms that define the amount of force or moment that isgenerated by each variable. These terms correspond to the fist-order terms of the Taylorseries expansion of the left-hand side variables with respect to each of the right-hand sideones. In addition, subscript 0 means constant biases in the force and moment equations.

The constant terms of the equations are initially set to zero, and their optimalvalues are obtained through Levenberg-Marquadt minimization of the adapted maximumlikelihood cost function (Equation 2.9). This aims to achieve the best fit between thereconstructed forces and moments and the ones generated by the linear polynomials fromEquation 2.11. This entire procedure is run through an adaptation of the OEM-Software.The minimization procedure is terminated after two iterations. Since the model is linear inparameters and in the independent variables, convergence is achieved within one iteration;however the software needs a second iteration to check if the minimum value has beenencountered.


The Two Step Method results in linear expressions for the total forces and momentsacting on the aircraft. However, if a linear state-space model of the overall aircraft isrequired, the 6DOF equations of motion have to be linearized as well. Figure 8 shows asimplified block diagram of a nonlinear aircraft simulation model. In the Two Step Method,a linear model for the dashed rectangle (total forces and moments) has been obtained;however the 6DOF equations of motion have been used in their nonlinear form during theidentification procedure.

Figure 8 – Simplified aircraft block diagram.

Source: The author.

In order to analyze the match between the measured states and the ones resultingfrom the Two Step Method, the longitudinal states have been generated by inputtingthe estimated forces and moments into the longitudinal 6DOF dynamic equations. Themeasurements of the lateral states are included, as well as the lateral angular ratesmeasurements corrected with the estimated biases, as shown in Equation 2.12.

u = X

m− qw + (rm −∆r)vm − g sin θ

w = Z

m− (pm −∆p)vm + qu− g cos θ cosφm

q = 1Iyy

(M +

((rm −∆r)2 − (pm −∆p)2

)Ixz + (pm −∆p)(rm −∆r)(Izz − Ixx)

)θ = q cosφm − (rm −∆r) sinφm

(2.12)where m is the aircraft mass; Iyy, Izz, Ixx, and Ixz denote the aircraft moments of inertiaand a product of inertia with respect to the axes indicated in the subscripts; X, Z, and


M are the estimated longitudinal forces and moments; subscript m stands for measuredlateral variables, and ∆ denotes the estimated biases in lateral angular rates.

2.5 Model validationModel validation, as defined by Schlesinger et al. (1974), is an essential process to

judge the applicability of the identified models to the intended purpose. It is performed bychecking whether the simulated and measured data match.

In order to provide means of comparing results with Lee (2017), the model validationperformed in this work has followed the same procedure presented by Lee.

The simulated states, obtained from both identification methods, have been evalu-ated by two statistical measures, namely Goodness of Fit (GOF) and Theil’s InequalityCoefficient (TIC). The former is a residual analysis criterion to measure model fit, whichcan be evaluated by multiple statistical metrics. In this project, the normalized meansquare error (NMSE) is used as GOF metric. Its definition is given in Equation 2.13.

Theil’s Inequality coefficient (Eq. 2.14), according to Jategaonkar (2015), providesmore insight about the correlation between measured and simulated data than the GOF.In the process of comparing estimates of dynamic system states, the NMSE measure canbe inadequate. As far as this metric concerned, even a straight line may be consideredbetter than a signal that is out of phase with the reference. According to Jategaonkar(2015), values of TIC<0.3 indicate good agreement.

NMSE varies between −∞ and 1, where negative values mean that a straight linematches the reference better than the estimated value, with −∞ as the worst possible fit,Positive values indicate that the estimate fits better than a straight line. A value of onemeans perfect match between the data. TIC values range between 0 and 1, and the lowerits values, the better the fit.

On the other hand, three statistical analyses have been applied in order to followwhat has been performed by Lee (2017) in the evaluation of the estimated forces andmoments. R2, root mean square error (RMSE), and normalized root mean squared error(NRMSE) are defined by Equations 2.15, 2.16, and 2.17, respectively. R2 varies between 0and 1. Zero indicates that the estimate fits worse than a straight line, and one means aperfect match.

RMSE is also known as the fit standard error. It has the same unit as the dependentvariable. Smaller values of this measure indicate better fit. NRMSE, on the other hand, isthe value RMSE divided by the variable range, which makes this measure dimensionless.


GOFi = NMSEi = 1− ||zi(tk)− yi(tk)||2

||zi(tk)− zi(tk)||2, i = 1, 2, . . . , ny (2.13)

TICi =

√1N

∑Nk=1[zi(tk)− yi(tk)]2√

1N

∑Nk=1[zi(tk)]2 +

√1N

∑Nk=1[yi(tk)]2

, i = 1, 2, . . . , ny (2.14)

R2i = 1−

∑Ni=1 [zi(tk)− yi(tk)]2∑Ni=1 [zi(tk)− zi(tk)]2

, i = 1, 2, . . . , ny (2.15)

RMSEi =√∑N

i=1[zi(tk)− yi(tk)]2N

, i = 1, 2, . . . , ny (2.16)

NRMSEi = RMSEizmax − zmin

, i = 1, 2, . . . , ny (2.17)

In Equations 2.13 through 2.17, N is the total number of data points, and ny is thetotal number of system outputs. z and y are the measurement and model output vectors,respectively. z is the the mean value of z, and ||.|| indicates the 2-norm of a vector.

A summary of the characteristics of the statistical measures used in this work ispresented in Table 1. The column named “Variables” stands for the variables evaluated byeach measure, according to what has been performed by Lee (2017). All measures can beinterchangeably used to evaluate states, forces, and moments. However, in order to allowa comparison with the results obtained by Lee (2017), the measures have been appliedin the same way. The column named “Threshold” defines the limits that indicate goodagreement between measurements and estimates. Except for TIC, a threshold for the othermeasures could not be found in the aircraft system identification literature.

Table 1 – Characteristics of the model validation statistic measures.

Measure Range Worst match Best match Threshold Variables (LEE, 2017)GOF (NMSE) −∞ ∼ 1 −∞ 1 undefined states

TIC 0 ∼ 1 1 0 <0.3 statesR2 −∞ ∼ 1 −∞ 1 undefined forces and moments

RMSE 0 ∼ +∞ +∞ 0 undefined forces and momentsNRMSE 0 ∼ 1 1 0 undefined forces and moments

Source: The author.

2.6 Parameter space control designParameter space control design is a widely used approach in robust control applica-

tions. It consists of designing a control system to meet some stability and/or performancespecifications by conducting a study on how the controlled plant will behave with respect toparameter variations (ACKERMANN, 1980). In robust control applications, both controland plant parameters are taken into account, which usually includes controller and model


variations, quantization effects, and sensor failures. The objective is to find a controllerthat behaves satisfactorily even if anomalous system variations occur. An application ofthis approach is described by Ackermann (2008), where the author cites his effort to finda set of fixed gains for an aircraft control system, instead of performing gain scheduling.

Ackermann (2012) and Lavretsky and Wise (2013) are books on robust control,which give a detailed explanation of the parameter space approach. Demirel and Guvenc(2010) and Saeki (2013) present cases where the technique has been applied.

The aim of this work slightly differs from robust control design objectives; however,a similar procedure is applied. The goal is to analyze the PID control parameter spacefor a set of predefined stability and performance requirements and check whether thecurrently used control gains fall inside or close to the intersection of the specificationswhen applied to the identified model. These gains have been tuned in flight using theZiegler-Nichols method, which does not require a model. Similar performance of the in-flight-tuned controller would be a possible indicator that the model is suitable for controldesign. This would eliminate the necessity of in-flight re-tuning and also allow for optimalcontrol design.

In order to perform that analysis, a grid search has been performed in a regionaround the currently used gains. Multiple combinations of proportional, integral, andderivative gains have been tested to check if the resulting closed-loop system would meetthe specifications.

For the purpose of this work, only the pitch attitude control system has beenanalyzed, whose control law is given in Equation 2.18.

δe = −(Kθ + Kiθ

s

)(θdes − θ)−Kq(qturn − q) (2.18)

where Kθ, Kiθ, and Kq are the proportional, integral and derivative gains, respectively. θand q are the measured aircraft pitch angle and angular rates. θdes is the reference for θand qturn is the reference for q, which is used for coordinated turns.

The requirements are: stable system, overshoot less than 5%, rise time (Tr) lessthan 1 second, and settling time (Ts), 5% criterion, less than 10 seconds. The result ofthe analysis is a set of plots. Each specification corresponds to a circle of a specific color.Circles are plotted in the coordinates corresponding to the parameters being tested, in casethe closed-loop system meets that requirement. A total of four plots have been generated,namely Kθ ×Kiθ, Kθ ×Kq, Kq ×Kiθ (Fig. 20), and a three-dimensional figure (Fig. 21).The gains that are currently being adopted and the range where the grid search has beenrun are shown in Table 2.

The ranges of gains have been chosen such that the behavior of the system couldbe analyzed in multiple points around the gains tuned in flight. The range of Kθ has been


Table 2 – Values tuned in flight and grid search range of gains.

Gain Currently adopted value Grid search rangeKθ 1 0.9 ∼ 1.2Kiθ 0.14 s−1 0.05 ∼ 0.4 s−1

Kq 0.2 s 0.1 ∼ 0.4 s

Source: The author.

defined not to cover gains much greater than 1. This value caused small occasional actuatorsaturations during the flight, which did not compromise the efficacy of the control system.The gains Kiθ and Kq have been initially set to range between 0.1 and 0.4 seconds−1, forthe former, and 0.1 and 0.4 seconds, for the latter. However, in order to find a minimumvalue of Kiθ that would not comply with the specifications for any values of the other twogains, the lower bound of its range has been slightly decreased to 0.05 seconds−1. The stepof variation for all gains has been set to 0.01.

27

3 RESULTS AND DISCUSSION

3.1 Considerations about the experimentThe natural frequencies of the a priori model are: 0.5358 rad/s and 9.0485 rad/s

(0.085 Hz and 1.44 Hz), which correspond to the long and short period modes, respectively.These frequencies have been used as inputs to Equations 2.3 and 2.4.

The characteristics of the optimal inputs, designed as described in Section 2.2, arepresented in Table 3. ∆t are the optimal time steps. These values have been calculated withEquations 2.3 and 2.4 for the throttle doublet and short period maneuvers, respectively,and chosen as described in Section 2.2 for the phugoid input. “Maximum amplitude” standsfor the highest amplitude of the input to avoid instability and actuator saturation, testedon the model presented by Balmer (2015). The maximum amplitude of the DLR 3211represents the average absolute amplitude of this signal. The throttle doublet input startsat throttle equal to 0.5, adds up to full throttle, decreases to zero, and returns to 0.5.“Oscillation time” is the minimum time that the aircraft should be allowed to oscillatefreely after the maneuver. This value is equivalent to the period of the mode it is intendedto excite (see Section 2.2).

For the purpose of this work, the bandwidth of the inputs have been considered tobe the range of frequencies that exhibit at least 50% of the maximum energy of the signal.It is assumed that only the dynamics within that range will be excited with sufficientenergy to affect the measurements. The analysis of identifiability of the derivatives iscarried out by taking into account the magnitude of their terms within the bandwidth ofthe signals.

Table 3 – Characteristics of optimal elevator and throttle inputs.

Maneuver ∆t(s) Maximum amplitude Oscillation time BandwidthShort period 0.18 0.6 rad 0.81 s 1.76 ∼ 15.60 rad/sPhugoid 1.6 0.18 rad 11.75 s 0 ∼ 1.73 rad/s

Throttle doublet 4.29 0.5 +− 0.5 11.75 s 0.27 ∼ 0.85 rad/sSource: The author.

The input signals in the time domain, as well as their power spectra are shown inFigure 9. The frequencies of the modes each maneuver is intended to excite are indicatedwith arrows.

Since there is no strict requirement on the time step of the long period input, avalue of ∆t equal to 1.6 seconds was found suitable. Longer time steps easily destabilizethe system. To avoid that, the elevator deflection would have to be reduced, which would

Chapter 3. RESULTS AND DISCUSSION 28

significantly lower the energy of the maneuver. Hence, the advantage of having largerenergy content in signals with longer ∆t described by Jategaonkar (2015) is, in this case,absent. On the other hand, small values of ∆t generate signals with energy contents spreadover a large range of frequencies, but with very small magnitudes, which may not beenough to properly excite the system (JATEGAONKAR, 2015). In that context, the timestep of the phugoid maneuver was chosen such that its bandwidth would not overlap withthe bandwidth associated with the short period maneuver.

(a) Elevator DLR 3211 in the time domain. (b) Elevator DLR 3211 power spectrum.

(c) Elevator pulse in the time domain. (d) Elevator pulse power spectrum.

(e) Throttle doublet in the time domain. (f) Throttle doublet power spectrum.

Figure 9 – Optimal inputs in the time and frequency domains.

Source: The author.

Due to time constraints, there has been no possibility of performing maneuverswith the optimal ∆t values found using Equations 2.3 and 2.4 in flight tests. Therefore, theavailable flight data has been used in system identification. The set of inputs performedin the flight tests has been designed by Lee (2017). Instead of centering the maximumenergy peak at the eigenfrequencies, Lee defined ∆t values such that the bandwidth of themaneuvers would be appropriate to identify all derivatives, except xδe, according to the10% rule. Regarding the number of parameters considered identifiable, both methods have


yielded the same results, despite the one presented by Lee (2017) not being concernedabout the optimality of the inputs.

Table 4 shows the characteristics of the actual maneuvers performed in flight.The input signals in the time domain, as well as their frequency responses are shown inFigure 10.

Table 4 – Characteristics of the inputs from flight data.

Maneuver ∆t(s) Amplitude Oscillation time BandwidthShort period 0.3 0.2 rad ∼7 s 1.04∼9.19 rad/sPhugoid 1 0.2 rad ∼27 s 0∼2.78 rad/s

Throttle doublet 3 0.6 +− 0.4 ∼27 s 0.38∼1.22 rad/sSource: Lee (2017).

(a) Elevator DLR 3211 in the time domain. (b) Elevator DLR 3211 power spectrum.

(c) Elevator pulse in the time domain. (d) Elevator pulse power spectrum.

(e) Throttle doublet in the time domain. (f) Throttle doublet power spectrum.

Figure 10 – Actual flight data inputs in the time and frequency domains.

Source: The author.

As previously discussed, the time step of the phugoid was chosen such that the inputdoes not excite frequencies already excited by the short period maneuver. However, since


the phugoid mode is lightly damped and requires relatively low energy to be excited, thechoice on ∆t is somewhat flexible. Hence, a time step of 1 second should yield equivalentresults to the one previously selected. In what concerns the throttle doublet maneuverapplied, its power spectrum has its peak at about 0.77 rad/s while the natural frequencyof the phugoid mode from the a priori model is excited with about 79% of the maximumpower. Despite not being optimal, this input may also be sufficient since the naturalfrequency of the mode is within the identifiable bandwidth of the maneuver and reasonablyclose to its peak. On the other hand, the DLR 3211 maneuver used in the flight test has itspeak at about 5.26 rad/s, while the natural frequency of the mode it is intended to excite islocated at a point that represents about 56% of the maximum power. Considering that thisvalue is still within the bandwidth, the maneuver is also considered acceptable. Adding tothat, it is known that the actual oscillation frequency of damped modes, i.e., their dampednatural frequency, is lower than their natural frequency (MILLER; MATTUCK, 2010).

After examining the time steps of the recorded inputs, a frequency domain analysisof the a priori model has been performed in order to validate the set of maneuvers, accordingto the 10% rule. To accomplish that, it is assumed that the linearized longitudinal modelof the aircraft is as given in Equation 3.1.

u = x0 + xuu+ xww + xqq + xθθ + xωω + xδeδe

w = z0 + zuu+ zww + zqq + zθθ + zδeδe

q = m0 +muu+mww +mqq +mωω +mδeδe

θ = q

ω = k0 + kuu+ kww −1Tωω + kω

Tωδt

(3.1)

where x0, z0, m0, and k0 are biases; xu, xw, xq, xθ, xω, xδe, zu, zw, zq, zθ, zδe, mu, mw,mq, mω, and mδe are the so called concise derivatives, which have this name becausethey represent larger expressions involving the aircraft dimensional derivatives. Theseare adopted for the sake of notation simplicity. The equivalent expressions to the concisederivatives can be found in Cook (2012). ku, kw, −1

Tω, kω

Tωare engine dynamics components.

The concise derivatives and engine dynamics components are defined in Appendix A.The engine dynamics equation has been assumed to be of this form in order to account forall nonzero components of the fifth line of the a priori model (Section 2.1). The Laplacetransform of this equation is shown in Appendix A. Furthermore, all the biases are assumedto be zero, and the values of the derivatives have been taken from the a priori model.

Bode plots containing each term on the right-hand side of Equation 3.1 have beengenerated in order to investigate the identifiability of the parameters within the rangeof each flight test maneuver, according to the 10% rule defined in Section 2.2. The plotscan be seen in Figure 11. The Bode diagrams of the derivative terms are generated by


plotting the frequency response magnitude of the terms in Equation 3.1 as a function ofthe input signal, which for the left-hand-side plots is the elevator command and for theright-hand-side figures is the throttle command.

To demonstrate the procedure applied, axial force equation is considered. Thefrequency response magnitudes of each of the terms, namely xu, xw, xq, xθ, xω, and xδe,are plotted with respect to the elevator input. The magnitudes being computed in thiscase are shown in Equation 3.2.

∣∣∣∣∣xuu(ω)δe(ω)

∣∣∣∣∣,∣∣∣∣∣xww(ω)δe(ω)

∣∣∣∣∣,∣∣∣∣∣xqq(ω)δe(ω)

∣∣∣∣∣,∣∣∣∣∣xθθ(ω)δe(ω)

∣∣∣∣∣,∣∣∣∣∣xωω(ω)δe(ω)

∣∣∣∣∣,∣∣∣∣∣xδeδe(ω)δe(ω)

∣∣∣∣∣ (3.2)

where argument (ω) denotes the Fourier transform of the variables.

The individual components are then computed from the output equation y =C[u w q θ ω]T + Dδe, with the observation matrices C and D defined as shown inEquation 3.3. The values of the vectors are taken from the a priori model. The subscriptsdenote the observation matrices for the corresponding components.

Cu = [−0.10 0 0 0 0], Du = [0]

Cw = [0 0.39 0 0 0], Dw = [0]

Cq = [0 0 − 1.4 0 0], Dq = [0]

Cθ = [0 0 0 − 9.8 0], Dθ = [0]

Cω = [0 0 0 0 0.006], Dω = [0]

Cδe = [0 0 0 0 0], Dδe = [0.38]

(3.3)

The same procedure has been performed for throttle as input and for all otherequations, except the one for θ since the first derivative of this variable is considered to beequal to q. Consequently, there are no parameters to be identified in this equation.

The blue, red and black dashed boxes shown in the Bode plots delimit the regionsof identifiability of phugoid, short period and throttle doublet maneuvers, respectively. Inaddition to the identifiability analysis, a study of the best input to identify each derivativeis also performed by analyzing the maximum magnitude of a term compared to the otherterms within the maneuver range. In order to avoid overlapping and make the choiceclearer, the left edge of the bandwidth of the short-period input was defined to start atthe right cutoff frequency of the phugoid maneuver. Since all frequencies in the bandwidthof the inputs are excited, the investigation is carried out by comparing the maximummagnitude of all terms within that range, even if the maximum values do not happen atthe same frequency. The results of this analysis are shown in Table 5.


(a) Axial force terms. Elevator as input. (b) Axial force terms. Throttle as input.

(c) Normal force terms. Elevator as input. (d) Normal force terms. Throttle as input.

(e) Pitching moment terms. Elevator as input. (f) Pitching moment terms. Throttle as input.

(g) Engine dynamics terms. Elevator as input. (h) Engine dynamics terms. Throttle as input.

Figure 11 – Bode plots of derivative terms. The blue, red, and black dashed rectanglesrepresent the regions of identifiability of the phugoid, short period, and throttledoublet maneuvers, respectively.

Source: The author.


In order to clarify the choice on the best maneuver to identify each derivative theanalysis of the mu term, from Figure 11 (e) and (f), is explained.

In the throttle doublet maneuver range, mu exhibits a maximum magnitude of 22.9dB, in contrast with the maximum values of the other terms, which are 21.7 dB, 2.54 dBand -1.91 dB.

In the bandwidth of the phugoid input, the maximum magnitude of mu is 42.5 dB,while the other terms have maximum peaks of 42.2 dB, 36.3 dB, 35.2 dB and 15.6 dB.

Analyzing the short period maneuver identifiability range, it can be seen that themu term has a maximum magnitude of -6.09 dB, while the other terms exhibit maximumvalues of 36.3 dB, 34.9 dB, 32.5 dB and -22.8 dB.

From this analysis it can be concluded that the term mu is best identified using thethrottle doublet maneuver since its relative maximum magnitude compared to the otherterms is greater than the ones exhibited in the ranges of the other two inputs. Besides thatthe term can be considered identifiable in the phugoid range and not identifiable withinthe short period bandwidth.

Furthermore, it can be noticed that the terms related to the parameters present inthe B matrix from the state-space model represent straight lines in the Bode diagrams. Thisoccurs due to the fact that these terms are static. It can be also noticed from Equation 3.2that the variables that multiply these parameters are canceled out.

Table 5 – Identifiability of derivatives.

Derivative Phugoid Short period Throttle doubletxu Best. Not identifiable. Identifiable.xw Identifiable. Best. Not identifiable.xq Identifiable. Best. Identifiable.xθ Identifiable. Identifiable. Best.xω Identifiable. Not identifiable. Best.xδe Not identifiable. Not identifiable. Not identifiable.zu Identifiable. Identifiable. Best.zw Identifiable. Best. Not identifiable.zq Identifiable. Best. Identifiable.zθ Best. Not identifiable. Identifiable.zδe Identifiable. Best. Not identifiable.mu Identifiable. Not identifiable. Best.mw Identifiable. Best. Not identifiable.mq Identifiable. Identifiable. Best.mω Best. Not identifiable. Identifiable.mδe Identifiable. Best. Not identifiable.ku Best. Identifiable. Identifiable.kw Identifiable. Best. Identifiable.− 1Tω

Identifiable. Best. Identifiable.kωTω

Not identifiable. Not identifiable. Best.Source: The author.

According to the examination of the magnitude of the terms, all parameters except


xδe can be identified using the flight data. It can be also concluded that the use of allkinds of maneuvers would possibly yield better results since the largest relative magnitudeof the terms are distributed over the three ranges. In addition, the use of different inputs isalso beneficial for the robustness of the parameter estimation, avoiding a possible limitedoperating range of the model.

From the available flight test data, three maneuvers of each kind have been chosenfor the identification procedure. Since an autopilot has been used, all inputs have beenadequately performed in terms of amplitude and time step; however in a few of them,enough time for the aircraft to oscillate has not been given. Those maneuvers have beendiscarded.

From the set of nine maneuvers considered useful, one of each class has been ran-domly chosen to compose the validation set, which has not been used during identification.Therefore, a total of six maneuvers have been used for identification and three maneuvershave been adopted for the validation procedure. Figures 12 and 13 display the inputs ofthe identification and validation sets, respectively. The vertical black lines delimit themaneuvers.

Figure 12 – Inputs used in identification.

Source: The author.


Figure 13 – Inputs used in validation.

Source: The author.

3.2 System identification results

3.2.1 Results of the a priori model

The states generated by the application of the validation maneuvers to the a prioristate-space model, presented in Section 2.1, are shown in Figure 14. The states simulatedwith the a priori model are plotted in red. The measured states are shown in blue.

Tables 6 and 7 present Goodness of Fit and Theil’s inequality coefficients, whichcompare the simulation results and the measurements of the states from the identificationand validation sets, respectively.

Table 6 – Statistical measures of the states simulated using the a priori model – identifi-cation set.

Measure u w q θ ωGOF -88.6034 -1.1008 -13.4752 -50.4809 -6.5016TIC 0.7384 0.7517 0.7793 0.8340 0.1149

Source: The author.


Figure 14 – States simulated with the a priori model and measured states.

Source: The author.

Table 7 – Statistical measures of the states simulated using the a priori model – validationset.

Measure u w q θ ωGOF -49.6545 -0.6371 -12.0559 -34.1344 -4.8069TIC 0.7466 0.6332 0.7864 0.8222 0.1309

Source: The author.

By analyzing Figure 14 and Tables 6 and 7 it can be concluded that the a priorimodel does not sufficiently simulate the aircraft behavior. Due to that, an effort to identifya new model by applying the Output Error and Two Step methods has been foundnecessary.

3.2.2 Results of the Output Error Method

The application of the Output Error Method resulted in the state-space modelshown in Equation 3.4. The state and measurement bias vectors are shown in Equation 3.5.Figure 15 presents a comparison between the time series of the measured and simulated


states. The latter are shown in red. These were generated through the integration of theleft-hand side of the state equation (Eq. 3.1), with the validation set of maneuvers (Fig. 13)as inputs. The measured states are plotted in blue.

u

w

q

θ

ω

=

−0.1122 1.143 −2.1000 −13.17 0.005679−0.6331 −2.860 22.02 −1.768 0−0.008642 −0.5316 −3.663 0 0.002205

0 0 1 0 05.401 −7.337 0 0 −2.480

u

w

q

θ

ω

+

0.38 0−37.13 0−27.81 0

0 00 1053

δeδt

(3.4)

bx =

−0.375616.3994−0.8661−0.05254655.4412

, by =

−1.12560.6713−0.05218

0.0093−35.3798

(3.5)

The GOF and TIC measures for both identification and validation sets are shownin Tables 8 and 9, respectively.

Table 8 – Statistical measures of the states resulting from the Output Error Method –identification set.

Measure u w q θ ωGOF 0.6604 0.3876 0.7225 0.7669 0.9363TIC 0.0347 0.2737 0.2861 0.2222 0.0102

Source: The author.

Table 9 – Statistical measures of the states resulting from the Output Error Method –validation set.

Measure u w q θ ωGOF 0.7366 -0.4452 0.6843 0.5399 0.9156TIC 0.0424 0.3514 0.3080 0.2879 0.0152

Source: The author.

The GOF values of w suggest poor matching between measurements and estimatesin both, identification and validation. The negative value in the latter suggests that astraight line would fit flight data better than the model prediction. However, the TICmeasure indicates that the match for this state, in validation, is reasonably close to the0.3 threshold, mentioned in Section 2.5, although being above it. A probable cause forthat is a possible inaccuracy of the GPS to acquire altitude variations.


Figure 15 – States simulated with the model resulting from the Output Error Method andmeasured states.

Source: The author.

In the validation set of θ, the TIC measure exhibits a value of 0.3080. This value isslightly higher than the limit of reasonable agreement between the data. Nevertheless, thisdeviation may not constrain control design.

The model prediction capacity for the other states is satisfactory, with ω exhibitingthe best fit. This may have occurred due to the fact that no scaling or normalization ispresent in the error covariance matrix, R, presented in Equation 2.8 or in the cost function,shown in Equation 2.9. This induces the optimization algorithm to minimize variablesthat exhibit higher values to the detriment of those with lower magnitudes, as long as thecost function is reduced.

An overall worse match in the throttle doublet maneuver for all states, except ω,may also indicate that adding the state ω to the state-space model, in order to accountfor the engine dynamics, does not exclude the necessity of having a proper throttle-thrustmapping if a closer fit is desired.

The identified state-space system has the following poles


Phugoid: -0.07331 +− 0.4256jShort period: -3.2166 +− 3.4474jEngine dynamics: -2.5354

Compared to the a priori model, the identified long and short period modes presentlower oscillation frequency. The former exhibits higher and the latter, lower damping. Theidentified engine dynamics time constant is slightly higher than the one previously found.

3.2.3 Results of the Two Step Method

This section is divided in two parts, which concern the presentation of the resultsof flight path reconstruction and the parameter estimation procedures, respectively.

3.2.3.1 Flight path reconstruction results

The first part of the Two Step Method yields an estimation of the biases present inthe linear acceleration and angular rate measurements. As described in Section 2.4.1, thevariables recorded in flight are used as inputs to the rigid-body 6DOF kinematic equations,shown in Equation 2.10. Then, an estimation of the bias values is carried out in order tominimize the errors between the time series of the measured states and the ones resultingfrom the integration of the 6DOF kinematic equations with biases included.

The predicted states from the validation set with biases resulting from maximumlikelihood estimation are plotted in Figure 16 together with the measurements. The valuesof the estimated biases for each maneuver are presented in Table 10.

Table 10 – Estimated measurement biases for each maneuver.

Bias Phugoid Short period Throttle doublet∆ax −9.409× 10−2 3.318× 10−2 2.650× 10−1

∆ay −1.511× 10−1 −3.422× 10−1 −2.666× 10−2

∆az −3.447× 10−3 −7.690× 10−2 −2.797× 10−2

∆p −3.233× 10−4 −3.070× 10−3 −8.556× 10−4

∆q −3.025× 10−4 3.437× 10−3 −4.018× 10−4

∆r 8.466× 10−3 5.642× 10−3 3.134× 10−3

Source: The author.

It can be noted that biases of the same nature presented different dimensions andeven different signs, which justifies the adoption of distinct estimates for each maneuver.

Table 11 shows the statistical measures of the conducted flight path reconstruction.

The overall procedure yielded satisfactory bias estimates. The match betweenmeasured and simulated vertical velocity, w, was again the poorest one; however, with aTIC of 0.20823, the agreement between the data is still considered reasonable, accordingto the criterion presented in Section 2.5.


Figure 16 – Measured longitudinal and lateral states and states generated from the inte-gration of the 6DOF equations of motion with estimated biases added.

Source: The author.

Table 11 – Statistical measures of the reconstructed states – validation set.

Measure u v w φ θ ψ hGOF 0.99879 0.93625 0.6556 0.8488 0.96966 0.99974 0.98777TIC 0.002799 0.11376 0.20823 0.18542 0.08097 0.007471 0.002550

Source: The author.

3.2.3.2 Results of force and moment components identification

Once acceptable values for biases had been found, an estimate of the actual forcesand moments acting in the aircraft during flight could be obtained and a linear regressionto find their composing terms has been performed.

The force and moment components, from the right-hand-side of Equation 2.11,estimated through this procedure are shown in Table 12. In Figure 17 the reconstructedforces and moments, as well as the ones resulting from the linear regression are shown.Tables 13 and 14 present the values of the statistical measures for the identification andvalidation sets, respectively.


Table 12 – Linear regression terms and respective values.

Term Value Term Value Term ValueX0 24.8912 Z0 -60.0449 M0 -1.6648Xu -1.9521 Zu -4.2001 Mu 0.06910Xw 1.9181 Zw -10.8758 Mw -0.4797Xq -1.5002 Zq -170.2218 Mq -7.4845Xδe -22.0493 Zδe 28.5036 Mδe -55.9172Xδt 44.7413

Source: The author.

Table 13 – Statistical measures of the Two Step Method estimated forces and moments –identification set.

Measure X Z MR2 0.7609 0.6232 0.4831

RMSE 4.5079 20.3579 0.9393NRMSE 0.08668 0.06067 0.04044

Source: The author.

Table 14 – Statistical measures of the Two Step Method estimated forces and moments –validation set.

Measure X Z MR2 0.7524 0.6593 0.03696

RMSE 4.6814 22.2496 1.2930NRMSE 0.09414 0.06072 0.05926

Source: The author.

The linear regression plots and statistics indicate that the procedure has yieldedacceptable fits, which means that, for this application, forces and moments can be satisfac-torily described by a linear combination of longitudinal states, presented in Equation 2.11.

Tables 15 and 16 present the statistical measures of the forces and momentsobtained by Lee (2017). As mentioned in Section 2.4, Lee has also performed Two Stepsystem identification for Penguin BE. Nonetheless, that author has performed flight pathreconstruction using an stochastic approach, based on an Extended Kalman Filter. In thesecond step, this work and Lee (2017) have followed the same procedure.

Table 15 – Statistical measures of the Two Step Method forces and moments obtained byLee (2017) – identification set.

Measure X Z MR2 0.868 0.616 0.422

RMSE 3.109 19.779 1.0899NRMSE 0.071 0.067 0.045

Source: Lee (2017).


Figure 17 – Reconstructed forces and moments, and forces and moments from linear-regression – validation set.

Source: The author.

Table 16 – Statistical measures of the Two Step Method forces and moments obtained byLee (2017) – validation set.

Measure X Z MR2 0.848 0.665 0.1359

RMSE 4.10 20.63 1.239NRMSE 0.078 0.067 0.053

Source: Lee (2017).

The statistical measures suggest comparable estimates of Z and M with the onesobtained by Lee (2017). Nevertheless, the estimation of X carried out by Lee (2017) seemto have yielded more reliable results.

The longitudinal states have been generated through the application of the estimatedforces and moments in the longitudinal 6DOF dynamic equations, with measurements


of the lateral states and the lateral angular rates corrected with the estimated biasesincluded, as shown in Equation 2.12.

The longitudinal states generated from the integration of the 6DOF dynamicequations, along with the measurements are plotted in Figure 18. The GOF and TICstatistical indicators of performance for the identification and validation sets are shown inTables 17 and 18, respectively.

Figure 18 – Measured and Two Step Method simulated states.

Source: The author.

Table 17 – Statistical measures of the states resulting from the Two Step Method – identi-fication set.

Measure u w q θGOF 0.3680 0.1027 0.5069 0.3745TIC 0.04729 0.3101 0.3423 0.3437

Source: The author.

The states w, q, and θ, generated by the integration of the 6DOF dynamic equations,present GOF measures below 0.5 in the validation set, which is possibly due to a frequencymismatch in the data from the throttle doublet maneuver. The TIC indicators for the


Table 18 – Statistical measures of the states resulting from the Two Step Method – valida-tion set.

Measure u w q θGOF 0.5330 -1.0559 0.4331 0.3207TIC 0.05612 0.3979 0.3833 0.3688

Source: The author.

same variables are above 0.3 for both identification and validation, which suggest deficientagreement between the data. Regarding the linear velocity u, TIC and GOF seem toindicate satisfactory estimation. Despite the latter not being much close to one, the formeris well below 0.3. The difference between the measurements and estimates of this state inthe first seconds may be due to a mismatch on the estimate of the aircraft axial force, X,within the same range.

Except by w, which has presented overall insufficient estimates in this technique,the other states seem to match the measured data appropriately for the phugoid andshort period maneuvers. The frequency mismatch and phase shift exhibited in the throttledoublet maneuver may possibly have occurred due to the fact that engine dynamics havenot been taken into account in the Two Step Method. A possible solution would be addingthe state ω to the linear expressions of the forces and moments, similarly to what hasbeen performed in the state equations in the Output Error Method. Another possibilitywould be estimating the aerodynamic forces and moments apart from the propulsion ones.

Tables 19 and 20 present the GOF and TIC measures for the states obtained byLee (2017). Identification and validation sets are shown.

Table 19 – Statistical measures of the Two Step Method states obtained by Lee (2017) –identification set.

Measure u w q θGOF 0.3680 0.1027 0.5069 0.3745TIC 0.04729 0.3101 0.3423 0.3437

Source: The author.

Table 20 – Statistical measures of the Two Step Method states obtained by Lee (2017) –validation set.

Measure u w q θGOF 0.5330 -1.0559 0.4331 0.3207TIC 0.05612 0.3979 0.3833 0.3688

Source: The author.

Except by the TIC values of q in the identification set and w in the validation setall other statistical measures suggest that the stochastic approach performed by Lee (2017)


has yielded better results than the deterministic one adopted in this work for the TwoStep Method.

3.2.4 Comparison between identification methods

Figure 19 shows measurements (in blue) and states resulting from the OutputError (in red) and Two Step (in green) methods from the validation set of maneuvers. Itis possible to note that estimates of w from the Output Error approach seem to matchthe flight data more closely than the the ones provided by the Two Step Method. Theother states have reasonably similar results in the first two maneuvers, with OEM yieldingslightly more accurate results in the third one.

Figure 19 – Measured States and states resulting from the Output Error and Two Stepidentified models.

Source: The author.

The statistical indicators suggest that the system identification performed usingthe Output Error Method has yielded overall better results than the Two Step Method. Apossible cause is the inclusion of the engine dynamics on the state-space model identifiedby the former. As mentioned in Section 3.2.3.2, adding the motor angular velocity, ω, to


the linear expressions of the forces and moments or identifying separate aerodynamic andpropulsion equations could possibly yield more adequate results for the Two Step Method.

The lack of a proper throttle-thrust mapping and the consequent adoption of thethrottle command as an input to the model has proven to be a jeopardizing factor for theachievement of better results. When engine dynamics are not included an instantaneousrelationship between throttle command and state variation is usually assumed. This causesthe effects of thrust variations in the model to instantly happen when the throttle commandis perturbed, which, on the actual aircraft, goes through a phase lag before it can benoticed by the sensors. The use of motor angular velocity ω seem to have slightly improvedthe match between the model and the real aircraft dynamics and can be considered asa solution in the absence of such mapping. Another possible reason to the mismatch,mentioned by Lee (2017), is that the fact that the states in the Output Error Method andthe forces and moments in the Two Step approach have only linear first order terms mayhave caused inaccurate estimates, especially in the throttle doublet maneuver.

Besides that, it is possible to note that the throttle doublet maneuver has generatedoscillations with larger amplitudes in θ. This may have caused the system to move out ofthe region of linearity. The maximum amplitude of the maneuvers that keep the systeminside that region has not been taken into account in the optimal input design process.However, in future experiments throttle doublets with lower amplitudes will be tested.

Furthermore, the non inclusion of elevator dynamics also appear to have causedsome phase shift and could be corrected by the addition of a mapping relating pilotcommand and actual surface deflection. Moreover, the estimates of w are not satisfactorilyaccurate, which may be due to possible GPS inaccuracies in measuring this state.

Adding to that, the fact that the Output Error Method has yielded an adequatelinear model, in contrast with the nonlinear results of the Two Step method from both, thiswork and Lee (2017), makes it a more attractive choice for control design. Linear equationsto represent the aircraft dynamics could also be generated by linearizing the modelsresulting from the Two Step Method. Nevertheless, no adequate linear representation couldbe found by applying such procedure. In fact, it seems to be a more reasonable choice toidentify the parameters of a linear model directly, instead of performing identification ofa nonlinear one and then linearizing it. The inaccuracies present in the nonlinear modelseem to be aggravated when the linearization procedure is carried out.

Although multiple techniques have been developed within aircraft nonlinear controlfield, like Backstepping (HARKEGARD; GLAD, 2007), those are mostly applied in case alinear model of the vehicle does not present adequate results. In this case, however, thelinear model identified using the Output Error Method will be utilized.


3.3 Results of Parameter Space Control DesignThe application of a step input to the system formed by the in-flight-tuned controller

and the identified model exhibited the following characteristics.

Table 21 – Performance indicators of the actual control system.

Overshoot Settling time Rise time1.69 % 9.87 s 0.97 s

Source: The author.

Figure 20 presents the results of the parameter space analysis of multiple controldesigns around the currently utilized gains, presented in Section 2.6. The figures showscatter plots of the values of the gains in combinations of two elements. In the left-side figures, the circles of different sizes and colors refer to the predefined stability andperformance specifications, namely stable system (yellow), overshoot less than 5% (blue),settling time (5% criterion) less than 10 seconds (green), and rise time less than 1 second(red). In these figures, a circle plotted at a pair of gains means that the system fulfillsthe specification represented by its color for those two gains and some value of the thirdgain, which is not included in the plot. In the figures on the right, black circles are plottedwhere the set of gains fulfilled all the requirements at the same time.

The divergence between the intersections of all colors on the left and the blackareas on the right-hand-side plots are due to the fact that, on the left plots, if circles ofmore than one color are present at a point, it is possible that the requirements have beenfulfilled for different values of the third gain that is not plotted. For instance, the overshootrequirement may have been fulfilled for Kθ = 1, Kiθ = 0.2 seconds−1, and Kq = 0.3seconds, and the settling time specification may have been met for Kθ = 1, Kiθ = 0.2seconds−1, and Kq = 0.1 seconds. In this situation, although the two requirements havebeen fulfilled for different values of Kq, a blue circle and a green one will be plotted at thepoint [1; 0.2] in the plot of Kθ ×Kiθ since it does not take the value of Kq into account.On the other hand, only the points where all the specifications have been met at the sametime are marked with a circle in the figures on the right.

The gains that are currently in use in flight experiments are represented on theleft and right-hand-side plots by black and green circles, respectively. Figure 21 is a three-dimensional plot of the region where all specifications are satisfied. It can be interpretedas a combination of the three figures on the right-hand side of Figure 20.

All sets of gains tested keep the system stable, although yellow circles cannot beclearly seem in some regions of the figures on the left.

It is possible to note that the gains currently used in the aircraft’s current autopilotfall inside the region formed by the specifications, which is a possible indicator that the


model is suitable for control design. Assuming that the identified plant perfectly matchesthe real one, any set of gains that fall inside the black regions could be chosen in order tomeet the requirements.

The parameter space analysis also gives an insight of which characteristics areimproved or worsened, if the gains are increased or decreased. In the left-side figures inFigure 20, it can be noted that decreasing Kiθ would increase the settling time. On theother hand, if the same gain is increased, the percentage of overshoot starts to get larger.From the same examination, one may conclude that excessively increasing Kq makes thesystem’s rise time exceed the maximum value specified. Small Kθ may also worsen settlingtime for some values of Kiθ. Nevertheless, actuator saturations that may occur for largevalues of that gain should also be accounted for. In this case, however, since the range ofKθ adopted is relatively small, saturation effects have not been taken into account.

From this analysis, other sets of gains with better performance could be chosen,in case the actual ones are unsatisfactory. An example would be Kθ = 1, Kiθ = 0.12seconds−1, and Kq = 0.18 seconds, which, when applied to the identified plant, exhibitsthe following characteristics: overshoot of 0.88%, settling time of 1.37 seconds, and risetime of 0.97 seconds.


Figure 20 – System performance in the parameter space. All specification regions on theleft. Intersection of specifications on the right. Left-hand-side plots: yellow –stable system; blue – overshoot less than 5%; green – settling time less than10 seconds; red – rise time less than 1 second; black – gains currently in use.Right-hand-side plots: black – gains where all specifications are met at thesame time; green – gains currently in use.

Source: The author.


Figure 21 – Intersection of specifications in the three-dimensional parameter space. Blackspheres – gains where all specifications are met at the same time; green sphere– gains currently in use

Source: The author.

51

4 CONCLUSION AND OUTLOOK

A reliable flight dynamics model for the Penguin BE aircraft used in cooperativelanding tests has been developed, with some inaccuracies in vertical velocity, w. An analysisof the pitch attitude control design that is currently in use has been performed, whichindicated similar performance of the control system applied to the actual system and itsidentified model. Furthermore, alternative control gains within the predefined specificationshave been suggested.

In order to test the suitability of the identified model, additional validation experi-ments shall be defined and executed in the near future. If those are successful, new controllaws shall be designed using the identified model. Once the techniques applied in thiswork have proven to yield adequate estimations, a lateral model shall also be identified. Inaddition, an engine model shall be obtained in order to reduce the mentioned inaccura-cies, especially in throttle variation maneuvers. Ultimately, a global system identificationprocedure shall be carried out. This would account for the aircraft dynamics within theentire flight envelope. Once all the techniques needed to obtain a high-fidelity model ofthe Penguin BE UAV are mastered, the first attempts to model flexible HALE aircraft forstratospheric missions shall be performed.

52

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Appendix

57

APPENDIX A – CONCISELONGITUDINAL AERODYNAMIC

DERIVATIVES AND ENGINE DYNAMICS

Table 22 shows the definition of the concise derivatives (COOK, 2012).

Table 22 – Definition of the concise derivatives.

Concise derivative Definitionxu Concise axial force derivative with respect to velocity along the x-axisxw Concise axial force derivative with respect to velocity along the z-axisxq Concise axial force derivative with respect to pitch ratexθ Concise axial force derivative with respect to pitch anglexω Concise axial force derivative with respect to motor angular velocityxδe Concise axial force derivative with respect to elevator commandzu Concise normal force derivative with respect to velocity along the x-axiszw Concise normal force derivative with respect to velocity along the z-axiszq Concise normal force derivative with respect to pitch ratezθ Concise normal force derivative with respect to pitch anglezδe Concise normal force derivative with respect to elevator commandmu Concise pitching moment derivative with respect to velocity along the x-axismw Concise pitching moment derivative with respect to velocity along the z-axismq Concise pitching moment derivative with respect to pitch ratemω Concise pitching moment derivative with respect to motor angular velocitymδe Concise pitching moment derivative with respect to elevator command

Source: The author.

Engine dynamics is assumed to follow Equation A.1.

ω(s) = kuu(s)s

+ kww(s)s

+ kω1 + Tωs

δt(s) (A.1)

where argument (s) indicates that variables are in frequency domain. ku, kw, and kω aregains, and Tω is the time constant of the system whose input is δt and output is ω.

This equation has been obtained by performing the Laplace transform of the fifthline of the a priori model, presented in Section 2.1.

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